Transfinite Induction

An extension of mathematical induction to well-ordered sets of ordinals: to prove a property P(α) for every ordinal α one shows P(0), proves P(α+1) assuming P(α) for each successor α+1, and proves P(λ) for each limit ordinal λ assuming P(β) for all β<λ.

An extension of mathematical induction to well-ordered sets indexed by ordinals: to prove a property P(α) for all ordinals α one shows P(0), proves that P(β) implies P(β+1) for any successor β+1, and for any limit ordinal λ shows that P(γ) holds for all γ < λ implies P(λ).